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Linear Algebra for Data Science


This is the 3rd post of blog post series ‘Data Science: The Complete Reference’, this post covers these topics related to data science introduction.

  1. What is Linear Algebra?

  2. Why Linear Algebra is important in Data Science?

  3. How Linear Algebra is applied in Data Science?

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What is Linear Algebra?

Linear algebra is the branch of mathematics concerning linear equations, linear functions and their representations through matrices and vector spaces. ~Wikipedia

https://ocw.mit.edu/courses/mathematics/18-700-linear-algebra-fall-2013/


The word algebra comes from the Arabi word “al-jabr” which means “the reunion of broken parts”. This is collection of methods deriving unknowns from knowns in mathematics. Linear Algebra is the branch that deals with linear equations and linear functions which are represented through matrices and vectors. In simpler words, it helps us understand geometric terms such as planes, in higher dimensions, and perform mathematical operations on them. By definition, algebra deals primarily with scalars (one-dimensional entities), but Linear Algebra has vectors and matrices (entities which possess two or more dimensional components) to deal with linear equations and functions.




Why Linear Algebra is significant in Data Science?

Linear Algebra is central to almost all areas of mathematics like geometry and functional analysis. Its concepts are a crucial prerequisite for understanding the theory behind Data Science. You don’t need to understand Linear Algebra before getting started in Data Science, but at some point, you may want to gain a better understanding of how the different algorithms really work under the hood. So if you really want to be a professional in this field, you will have to master the parts of Linear Algebra that are important for Data Science.


How Linear Algebra is used in Data Science?

Scalars, Vectors, Matrices and Tensors

  1. A scalar is a single number

  2. A vector is an array of numbers.

  3. A matrix is a 2-D array

  4. A tensor is a n-dimensional array with n>2


With transposition we can convert a row vector to a column vector and vice versa.

Matrix Transposition



Multiplying Matrices and Vectors

The dot product of matrices & vectors is used in every equation explaining data science algorithms. Matrices multiplication is distributive, associative, NOT commutative, while vector multiplication is commutative.

Matrix Multiplication



Identity and Inverse Matrices

The identity matrix In is a special matrix of shape (n×n) that is filled with 0 except the diagonal that is filled with 1. An inverse matrix is that results in the identity matrix when it is multiplied by its original form.

Identity Matrix


Inverse Matrix



Linear Dependence and Span

We cover here how to represent systems of equations graphically, how to interpret the number of solutions of a system, what is linear combination, dependence and span.


http://algebra.math.ust.hk/vector_space/06_span/lecture2.shtml



Norms

The norm is what is generally used to evaluate the error of a model. For instance it is used to calculate the error between the output of a neural network and what is expected (the actual label or value).

The squared L2 norm



Special Kinds of Matrices and Vectors

This section covers different interesting type of matrices with specific properties i.e. Diagonal, Symmetric & Orthogonal matrices.




Eigendecomposition

The eigendecomposition is one form of matrix decomposition. Decomposing a matrix means that we want to find a product of matrices that is equal to the initial matrix. In the case of the eigendecomposition, we decompose the initial matrix into the product of its eigenvectors and eigenvalues.

The eigendecomposition of the matrix corresponding to a quadratic equation can be used to find the minimum and maximum of that function.


Singular Value Decomposition (SVD)

The way to go to decompose other types of matrices that can’t be decomposed with eigendecomposition is to use SVD. With the SVD, we decompose a matrix in three other matrices. We can see these new matrices as sub-transformations of the space. Instead of doing the transformation in one movement, we decompose it in three movements.



The Moore Penrose Pseudoinverse

The inverse is used to solve system of equations but not all matrices have an inverse. In some cases, a system of equation has no solution, and thus the inverse doesn’t exist. However it can be useful to find a value that is almost a solution (in term of minimizing the error). We can find the best-fit line of a set of data points with the pseudoinverse.


The Trace Operator

The trace is the sum of all values in the diagonal of a square matrix. It can be used to specify the Frobenius norm of a matrix, which is needed for the Principal Component Analysis (PCA)



The Determinant

The determinant of a matrix AA is a number corresponding to the multiplicative change you get when you transform your space with this matrix. A negative determinant means that there is a change in orientation (and not just a rescaling and/or a rotation).



Principal Component Analysis

When the data-set is high-dimensional, it would be nice to have a way to reduce these dimensions while keeping all the information present in the data-set. The aim of principal components analysis (PCA) is generally to reduce the number of dimensions of a data-set where dimensions are not completely decorrelated.



References




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